By updating on May 25, 2022, the content got to be able to be embedded as a full page in the new Google site. Ref. This is very good news for me.
For example, with this update, the Web Apps created by Google Apps Script can be published by embedding to the Google site as a full page. By this, the URL of the Google site, and Google Analytics can be used. I have already changed my Web Applications. You can see them at https://tanaikech.github.io/webapps/
This sample gif image is from “Embed content as a full page in new Google Sites”.